May 2012 - Banach - mathdept.okstate.edu

Abstract of a paper by Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Greedy algorithms for reduced bases in Banach spaces" by Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk. Abstract: Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n dimensional space X_n ⊂ X which can be used to approximate the elements of F. The best possible error we can achieve for such an approximation is given by the Kolmogorov width d_n(F)_X. However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in A. Buffa, Y. Maday, A.T. Patera, C. Prud’homme, and G. Turinici, ''A Priori convergence of the greedy algorithm for the parameterized reduced basis'', M2AN Math. Model. Numer. Anal., 46(2012), 595–603 in the case X = H is a Hilbert space. The results there were significantly improved on in P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk, ''Convergence rates for greedy algorithms in reduced bases Methods'', SIAM J. Math. Anal., 43 (2011), 1457–1472. The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces. Archive classification: math.FA Mathematics Subject Classification: 41A46, 41A25, 46B20, 15A15 Submitted from: gpetrova(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2290 or http://arXiv.org/abs/1204.2290

1 0

Abstract of a paper by G. Botelho, D. Cariello, V.V. Favaro, D. Pellegrino and J.B. Seoane-Sepulveda
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Subspaces of maximal dimension contained in $L_{p}(\Omega) - \textstyle\bigcup\limits_{q<p}L_{q}(\Omega)$}" by G. Botelho, D. Cariello, V.V. Favaro, D. Pellegrino and J.B. Seoane-Sepulveda. Abstract: Let $(\Omega,\Sigma,\mu)$ be a measure space and $1< p < +\infty$. In this paper we determine when the set $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$ is maximal spaceable, that is, when it contains (except for the null vector) a closed subspace $F$ of $L_{p}(\Omega)$ such that $\dim(F) = \dim\left(L_{p}(\Omega)\right)$. The aim of the results presented here is, among others, to generalize all the previous work (since the 1960's) related to the linear structure of the sets $L_{p}(\Omega) - L_{q}(\Omega)$ with $q < p$ and $L_{p}(\Omega) - \bigcup\limits_{1 \leq q < p}L_{q}(\Omega)$. We shall also give examples, propose open questions and provide new directions in the study of maximal subspaces of classical measure spaces. Archive classification: math.FA Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2170 or http://arXiv.org/abs/1204.2170

1 0

Abstract of a paper by Jan-David Hardtke
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "A remark on condensation of singularities" by Jan-David Hardtke. Abstract: Recently Alan D. Sokal gave a very short and completely elementary proof of the uniform boundedness principle. The aim of this note is to point out that by using a similiar technique one can give a considerably short and simple proof of a stronger statement, namely a principle of condensation of singularities for certain double-sequences of non-linear operators on quasi-Banach spaces, which is a bit more general than a result of I.\,S. G\'al. Archive classification: math.FA Mathematics Subject Classification: 46A16, 47H99 Remarks: 7 pages Submitted from: hardtke(a)math.fu-berlin.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2106 or http://arXiv.org/abs/1204.2106

1 0

Abstract of a paper by Jean-Matthieu Auge
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Perturbation of farthest points in weakly compact sets" by Jean-Matthieu Auge. Abstract: If $f$ is a real valued weakly lower semi-continous function on a Banach space $X$ and $C$ a weakly compact subset of $X$, we show that the set of $x \in X$ such that $z \mapsto \|x-z\|-f(z)$ attains its supremum on $C$ is dense in $X$. We also construct a counter example showing that the set of $x \in X$ such that $z \mapsto \|x-z\|+\|z\|$ attains its supremum on $C$ is not always dense in $X$. Archive classification: math.FA Mathematics Subject Classification: Primary 41A65 Remarks: 5 pages Submitted from: jean-matthieu.auge(a)math.u-bordeaux1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2047 or http://arXiv.org/abs/1204.2047

1 0

Abstract of a paper by Jean-Matthieu Auge
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Orbits of linear operators and Banach space geometry" by Jean-Matthieu Auge. Abstract: Let $T$ be a bounded linear operator on a (real or complex) Banach space $X$. If $(a_n)$ is a sequence of non-negative numbers tending to 0. Then, the set of $x \in X$ such that $\|T^nx\| \geqslant a_n \|T^n\|$ for infinitely many $n$'s has a complement which is both $\sigma$-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents $q>0$, such that for every non nilpotent operator $T$, there exists $x \in X$ such that $(\|T^nx\|/\|T^n\|) \notin \ell^{q}(\mathbb{N})$, using techniques which involve the modulus of asymptotic uniform smoothness of $X$. Archive classification: math.FA Mathematics Subject Classification: Primary 47A05, 47A16, Secondary 28A05 Remarks: 16 pages Submitted from: jean-matthieu.auge(a)math.u-bordeaux1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2046 or http://arXiv.org/abs/1204.2046

1 0

Abstract of a paper by Jean-Matthieu Auge
by alspach＠math.okstate.edu 08 May '12

08 May '12

This is an announcement for the paper "Linear operators with wild dynamics" by Jean-Matthieu Auge. Abstract: If $X$ is a separable infinite dimensional Banach space, we construct a bounded and linear operator $R$ on $X$ such that $$ A_R=\{x \in X, \|R^tx\| \rightarrow \infty\} $$ is not dense and has non empty interior with the additional property that $R$ can be written $I+K$, where $I$ is the identity and $K$ is a compact operator. This answers two recent questions of H\'ajek and Smith. Archive classification: math.FA Mathematics Subject Classification: Primary 47A05, Secondary 47A15, 47A16 Remarks: 14 pages Submitted from: jean-matthieu.auge(a)math.u-bordeaux1.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1204.2044 or http://arXiv.org/abs/1204.2044

1 0